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We investigate the critical properties of continuous random field Ising model (RFIM). Using the distributional zeta-function method, we obtain a series representation for the quenched free energy. It is possible to show that for each moment of the partition function, the multiplet of $k$-fields the Gaussian contribution has one field with the contribution of the disorder and $(k-1)$-fields with the usual propagator. Although the non-gaussian contribution is non-perturbative we are able to show that the model is confined between two $\mathbb{Z}_2\times\mathcal{O}(k-1)$-symmetric models. Using arguments of lower critical dimension alongside with monotone operators, we show that the phase of the continuous RFIM can be restricted by an $\mathbb{Z}_2 \times \mathcal{O}(k-1) \to \mathcal{O}(k-2)$ phase transition., Comment: 6 pages

The Random Sequential Adsorption (RSA) problem holds crucial theoretical and practical significance, serving as a pivotal framework for understanding and optimizing particle packing in various scientific and technological applications. Here the problem of the one-dimensional RSA of k-mers onto a substrate with correlated defects controlled by uniform and power-law distributions is theoretically investigated: the coverage fraction is obtained as a function of the density of defects and several scaling laws are examined. The results are compared with extensive Monte Carlo simulations and more traditional methods based on master equations. Emphasis is given in elucidating the scaling behavior of the fluctuations of the coverage fraction. The phenomenon of universality breaking and the issues of conventional gaussian fluctuations and the L\'evy type fluctuations from a simple perspective, relying on the Central Limit Theorem, are also addressed.

We propose a new approach based on random matrix theory to calculate the temporal second-order intensity correlation function $g^{(2)}(t)$ of the radiation emitted by random lasers and random fiber lasers. The multimode character of these systems, with a relevant degree of disorder in the active medium, and large number of random scattering centers substantially hinder the calculation of $g^{(2)}(t)$. Here we apply for the first time in a photonic system the universal statistical properties of Ginibre's non-Hermitian random matrix ensemble to obtain $g^{(2)}(t)$. Excellent agreement is found with time-resolved measurements for several excitation powers of an erbium-based random fiber laser. We also discuss the extension of the random matrix approach to address the statistical properties of general disordered photonic systems with various Hamiltonian symmetries., Comment: 5 pages, 4 figures

We offer a new perspective to the problem of characterizing mesoscopic fluctuations in the inter-plateau region of the integer quantum Hall transition. We found that longitudinal and transverse conductance fluctuations, generated by varying the external magnetic field within a microscopic model, are multifractal and lead to distributions of conductance increments (magnetoconductance) with heavy tails (intermittency) and signatures of a hierarchical structure (a cascade) in the corresponding stochastic process, akin to Kolmogorov's theory of fluid turbulence. We confirm this picture by interpreting the stochastic process of the conductance increments in the framework of H-theory, which is a continuous-time stochastic approach that incorporates the basic features of Kolmogorov's theory. The multifractal analysis of the conductance "time series," combined with the H-theory formalism provides, strong support for the overall characterization of mesoscopic fluctuations in the quantum Hall transition as a multifractal stochastic phenomenon with multiscale hierarchy, intermittency, and cascade effects.

We perform a multifractal detrended fluctuation analysis of the magnetoconductance data of two standard types of mesoscopic systems: a disordered nanowire and a ballistic chaotic billiard, with two different lattice structures. We observe in all cases that multifractality is generally present and that it becomes stronger in the quantum regime of conduction, i.e., when the number of open scattering channels is small. We argue that this behavior originates from correlations induced by the magnetic field, which can be characterized through the distribution of conductance increments in the corresponding "stochastic time series," with the magnetic field playing the role of a fictitious time. More specifically, we show that the distributions of conductance increments are well fitted by $q$ Gaussians and that the value of the parameter $q$ is a useful quantitative measure of multifractality in magnetoconductance fluctuations.

Skewness and non-Gaussian behavior are essential features of the distribution of short-scale velocity increments in isotropic turbulent flows. Yet, although the skewness has been generally linked to time-reversal symmetry breaking and vortex stretching, the form of the asymmetric heavy tails remain elusive. Here we describe the emergence of both properties through an exactly solvable stochastic model with a scale hierarchy of energy transfer rates. From a statistical superposition of a local equilibrium distribution weighted by a background density, the increments distribution is given by a novel class of skewed heavy-tailed distributions, written as a generalization of the Meijer $G$-functions. Excellent agreement in the multiscale scenario is found with numerical data of systems with different sizes and Reynolds numbers. Remarkably, the single scale limit provides poor fits to the background density, highlighting the central role of the multiscale mechanism. Our framework can be also applied to describe the challenging emergence of skewed distributions in complex systems., Comment: 25 pages, 5 figures. To appear in Physical Review Fluids, 2019

We analyze the heat exchange distribution of quantum open systems undergoing a thermal relaxation that maximizes the entropy production. We show that the process implies a type of generalized law of cooling in terms of a time dependent effective temperature $T_t$. Using a two-point measurement scheme, we find an expression for the heat moment generating function that depends solely on the system's partition function and on the law of cooling. Applications include the relaxation of free bosonic and fermionic modes, for which closed form expressions for the time-dependent heat distribution function are derived. Multiple free modes with arbitrary dispersion relations are also briefly discussed. In the semiclassical limit our formula agrees well with previous results of the literature for the heat distribution of an optically trapped nanoscopic particle far from equilibrium., Comment: 6 pages, 2 figures

Turbulence is a challenging feature common to a wide range of complex phenomena. Random fibre lasers are a special class of lasers in which the feedback arises from multiple scattering in a one-dimensional disordered cavity-less medium. Here, we report on statistical signatures of turbulence in the distribution of intensity fluctuations in a continuous-wave-pumped erbium-based random fibre laser, with random Bragg grating scatterers. The distribution of intensity fluctuations in an extensive data set exhibits three qualitatively distinct behaviours: a Gaussian regime below threshold, a mixture of two distributions with exponentially decaying tails near the threshold, and a mixture of distributions with stretched-exponential tails above threshold. All distributions are well described by a hierarchical stochastic model that incorporates Kolmogorov's theory of turbulence, which includes energy cascade and the intermittence phenomenon. Our findings have implications for explaining the remarkably challenging turbulent behaviour in photonics, using a random fibre laser as the experimental platform., Comment: 9 pages, 5 figures

A novel formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem---representing the region where the measurements are made---in contact with a set of `nested heat reservoirs' corresponding to the hierarchical structure of the system. The probability distribution function (pdf) of the fluctuating temperatures at each reservoir, conditioned on the temperature of the reservoir above it, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox $H$-functions. The distribution of states of the small subsystem is then computed by averaging the quasi-equilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of $H$-functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Mac\^edo {\it et al.} [Phys.~Rev.~E {\bf 95}, 032315 (2017)] from a stochastic dynamical approach to the problem., Comment: 20 pages, 2 figures

A unified approach is proposed to describe the statistics of the short time dynamics of multiscale complex systems. The probability density function of the relevant time series (signal) is represented as a statistical superposition of a large time-scale distribution weighted by the distribution of certain internal variables that characterize the slowly changing background. The dynamics of the background is formulated as a hierarchical stochastic model whose form is derived from simple physical constraints, which in turn restrict the dynamics to only two possible classes. The probability distributions of both the signal and the background have simple representations in terms of Meijer G-functions. The two universality classes for the background dynamics manifest themselves in the signal distribution as two types of tails: power law and stretched exponential, respectively. A detailed analysis of empirical data from classical turbulence and financial markets shows excellent agreement with the theory., Comment: 6 pages, 4 figures

Quantum transport through devices coupled to electron reservoirs can be described in terms of the full counting statistics (FCS) of charge transfer. Transport observables, such as conductance and shot-noise power are just cumulants of FCS and can be obtained from the sample's average density of transmission eigenvalues, which in turn can be obtained from a finite element representation of the saddle-point equation of the Keldysh (or supersymmetric) non-linear sigma-model, known as quantum circuit theory. Normal universal metallic behavior in the semiclassical regime is controlled by the presence of a Fabry-Perot singularity in the average density of transmission eigenvalues. We present general conditions for the suppression of Fabry-Perot modes in the semiclassical regime in a sample of arbitrary shape, a disordered conductor or a network of ballistic quantum dots, which leads to an anomalous metallic phase. Through a double-scaling limit, we derive a scaling equation for anomalous metallic transport, in the form of a nonlinear differential equation, which generalizes the ballistic-diffusive scaling equation of a normal metal. The two-parameter stationary solution of our scaling equation generalizes Dorokhov's universal single-parameter distribution of transmission eigenvalues. We provide a simple interpretation of the stationary solution using a thermodynamic analogy with a spin-glass system. As an application, we consider a system formed by a diffusive wire coupled via a barrier to normal-superconductor (NS) reservoirs. We observe anomalous reflectionless tunneling, when all perfectly transmitting channels are suppressed, which cannot be explained by the usual mechanism of disorder-induced opening of tunneling channels., Comment: 24 pages, 4 figures

We derive a set of coupled non-linear algebraic equations for the asymptotics of the Poisson kernel distribution describing the statistical properties of a two-terminal double-barrier chaotic billiard (or ballistic quantum dot). The equations are calculated from a diagrammatic technique for performing averages over the unitary group, proposed by Brouwer and Beenakker [J. Math. Phys. 37, 4904 (1996)]. We give strong analytical evidences that these equations are equivalent to a much simpler polynomial equation calculated from a recent extension of Nazarov's circuit theory [A. M. S. Macedo, Phys. Rev. B 66, 033306 (2002)]. These results offer interesting perspectives for further developments in the field via the direct conversion of one approach into the other., Comment: 7 pages, to appear in Physical Review B

We present an exact solution of a supersymmetric nonlinear sigma model describing the crossover between a quantum dot and a disordered quantum wire with unitary symmetry. The system is coupled ideally to two electron reservoirs via perfectly conducting leads sustaining an arbitrary number of propagating channels. We obtain closed expressions for the first three moments of the conductance, the average shot-noise power and the average density of transmission eigenvalues. The results are complete in the sense that they are nonperturbative and are valid in all regimes and length scales. We recover several known results of the recent literature by taking particular limits., Comment: 4 pages

A Brownian motion model is proposed to study parametric correlations in the transmission eigenvalues of open ballistic cavities. We find interesting universal properties when the eigenvalues are rescaled at the hard edge of the spectrum. We derive a formula for the power spectrum of the fluctuations of transport observables as a response to an external adiabatic perturbation. Our formula correctly recovers the Lorentzian-squared behaviour obtained by semiclassical approaches for the correlation function of conductance fluctuations., Comment: 19 pages, written in RevTeX

We obtain explicit expressions for the correlation functions of transmission and reflection coefficients of coherent electronic waves propagating through a disordered quasi-one-dimensional medium with purely elastic diffusive scattering in the presence of spin-orbit interactions. We find in the metallic regime both large local intensity fluctuations and long-range correlations which ultimately lead to universal conductance fluctuations. We show that the main effect of spin-orbit scattering is to suppress both local and long-range intensity fluctuations by a universal symmetry factor 4. We use a scattering approach based on random transfer matrices., Comment: 15 pages, written in plain TeX, Preprint OUTP-93-42S (University of Oxford), to appear in Phys. Rev. B

We study the response of the transmission eigenvalue spectrum of disordered metallic conductors to an arbitrary external perturbation. For systems without time-reversal symmetry we find an exact non-perturbative solution for the two-point correlation function, which exhibits a new kind of universal behavior characteristic of disordered conductors. Systems with orthogonal and symplectic symmetries are studied in the hydrodynamic regime., Comment: 10 pages, written in plain TeX, Preprint OUTP-93-36S (University of Oxford), to appear in Phys. Rev. B (Rapid Communication)

We extend a recent theory of parametric correlations in the spectrum of random matrices to study the response to an external perturbation of eigenvalues near the soft edge of the support. We demonstrate by explicit non-perturbative calculation that the two-point function for level density fluctuations becomes, after appropriate rescaling, a universal expression., Comment: 8 pages, written in TeX, Preprint OUTP-94-10S (University of Oxford)

We study the effects of an arbitrary external perturbation in the statistical properties of the S-matrix of quantum chaotic scattering systems in the limit of isolated resonances. We derive, using supersymmetry, an exact non-perturbative expression for the parameter dependent autocorrelator of two S-matrix elements. Universality is obtained by appropriate rescaling of the physical parameters. We propose this universal function as a new signature of quantum chaos in open systems., Comment: 4 pages, 1 figure appended, written in REVTeX, Preprint OUTP-94-13S (University of Oxford)